Certification Problem

Input (COPS 980)

We consider the TRS containing the following rules:

a(x)	→	b(x)	(1)
b(b(c(x)))	→	c(a(c(a(a(x)))))	(2)
c(c(x))	→	x	(3)

The underlying signature is as follows:

{a/1, b/1, c/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t₀	=	b(b(c(c(x109))))
	→	b(b(x109))
	=	t₁

t₀	=	b(b(c(c(x109))))
	→	c(a(c(a(a(c(x109))))))
	→	c(a(c(a(b(c(x109))))))
	→	c(a(c(b(b(c(x109))))))
	→	c(b(c(b(b(c(x109))))))
	→	c(b(c(c(a(c(a(a(x109))))))))
	→	c(b(c(c(a(c(a(b(x109))))))))
	→	c(b(c(c(a(c(b(b(x109))))))))
	→	c(b(c(c(b(c(b(b(x109))))))))
	→	c(b(b(c(b(b(x109))))))
	→	c(c(a(c(a(a(b(b(x109))))))))
	→	c(c(a(c(a(b(b(b(x109))))))))
	→	c(c(a(c(b(b(b(b(x109))))))))
	→	c(c(b(c(b(b(b(b(x109))))))))
	→	b(c(b(b(b(b(x109))))))
	=	t₁₄

The two resulting terms cannot be joined for the following reason:

When applying the cap-function on both terms (where variables may be treated like constants) then the resulting terms do not unify.